PROBLEMS ON TRAINS WITH SOLUTIONS - 1

Problem 1 :

Two stations A and B are 110 km apart on a straight line. One train starts from A at 7 a.m. and travels towards B at 20 kmph. Another train starts from B at 8 a.m. and travels towards A at a speed of 25 kmph. At what time will they meet ?

Solution :

Let the trains meet each other "m" hours after 7 a.m. 

Distance covered by the train from A in "m" hrs is

=  Speed ⋅ Time

=  20 kms

At a particular time after 8 a.m, if the train from A had traveled "m" hours, then the train from B would have traveled (m-1) hours.

Because it started at 8.00 am. (one hour later)

Distance covered by the train from B in (m - 1) hrs  is

=  25(m - 1) 

At the meeting point, 

sum of the distances covered by the two trains is equal to the total distance (from A to B). 

That is

20m + 25(m-1)  =  110

20m + 25m - 25  =  110 

45m  =  135

  m  =  3 hours

So, two trains meet each other 3 hrs after 7 a.m

That is, at 10 a.m. 

Hence, the time at which they will meet is 10 a.m.

Problem 2 :

Two trains are running at 40 kmph and 20 kmph respectively in the same direction .Faster train completely passes a man who is sitting in the slower train in 9 seconds. What is the length of the faster train?

Solution :

Relative speed of two trains is 

=  40 - 20  

=  20 kmph

=  20 ⋅ 5/18 m/sec  

=  50/9 m/sec 

Given : Faster train completely passes a man who is sitting in the slower train in 9 seconds.

Length of the faster train is

=  The distance covered by the faster train in this 9 seconds  

=  Speed ⋅ Time

=  50/9 ⋅ 9

=  50 m 

Hence, the length of the faster train is 50 m.

Problem 3 :

Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. Find the ratio of their speeds 

Solution :

Let "a" m/sec and "b" m/sec be the speeds of two trains respectively 

First train crosses the man in 27 seconds with speed "a" m/sec

So, the length of the first train is

=  Distance covered in 27 seconds

=  Speed ⋅ Time 

=  a ⋅ 27

=  27a -----(1)

Second train crosses the man in 17 seconds with speed "b" m/sec

Length of the second train is 

=  Distance covered in 17 seconds

=  Speed ⋅ Time

=  b ⋅ 17

=  17b -----(2)

Given : The given two trains cross each other in 23 seconds. 

The distance covered by the two trains in this 23 seconds is 

 =  Sum of the lengths of the two trains 

=  Relative speed ⋅ Time

=  (a + b) ⋅ 23

=  23a + 23b -----(3)

We know the fact that when two trains cross each other in opposite directions, the distance covered by them is equal to sum of the length of the two trains.  

That is, 

(1) + (2)  =  (3)

27a + 17b  =  23a + 23b

4a  =  6b 

a / b  =  6 / 4

a / b  =  3 / 2

a : b  =  3 : 2

Hence, the ratio of their speeds is 3:2. 

Problem 4 :

A train passes a station platform in 36 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 54 km/hr, what is the length of the platform ?

Solution :

Speed of the train  =  54 kmph

=  54 ⋅ 5/18 m/sec

=  15 m/sec 

The train passes the man in 20 seconds. 

The distance covered by the train in this 20 seconds is equal to the length of the train. 

Distance  =  Speed ⋅ Time

Distance  =  20 ⋅ 15

Distance  =  300 m 

So, length of the train is 300 m.  

Let "m" be the length of the platform 

Given : The train crosses the  platform in 36 seconds.  

We know the fact that the distance covered by the train in this 36 seconds is equal to sum of the lengths of the train and platform 

Then, the distance covered by the train in 36 seconds is

 =  300 + m 

So, the train takes 36 seconds to cover the distance "300 + m"

Time  =  Distance / Speed 

36  =  (300 + m) / 15 

540  =  300 + m 

240  =  m  

Hence, the length of the platform is 240 meters.

Problem 5 :

Two trains are moving in opposite directions at 60 km/hr and 90 km/hr. Their lengths are 1.10 km and 0.9 km respectively. Find the time taken by the two trains to cross each other.

Solution :

Relative speed  is

=  60 + 90  =  150 kmphr

=  150 ⋅ 5/18 m/sec

=  125/3 m/sec 

When they cross each other, distance covered by both the trains  is equal to sum of the lengths of the two trains.  

So, the distance covered by them is

=  1.1 + 0.9  

=  2 km

=  2 ⋅ 1000 m  

=  2000 m 

Time taken by the two trains to cross each other is 

=  Distance / Speed 

=  2000 / (125/3)

=  2000 ⋅ 3/125

=  48 seconds 

Hence, time taken by the two trains to cross each other is 48 seconds.

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